Paradox regarding the size of real numbers?

Question

Suppose you want to represent numbers in base 9, so digits 0 to 8. You can express any real number in base 9, so this set is uncountably infinite.

However, it can be shown that in base 10, the likelihood of any real number containing a 9 approaches 100%, meaning that the numbers not containing 9 are insignificant, approaching 0%, to those that do. E.g. similar to this proof using 7, https://www.flyingcoloursmaths.co.uk/percentage-numbers-seven/

However, this set of numbers not containing 9 is identical to those numbers expressed in base 9. So we have an uncountable infinite set (real numbers in base 9) being 0% of the size of another uncountable infinite set (real numbers in base 10).

This can continue, real numbers in base 8 vs real numbers in base 9, real numbers in base 7 vs real numbers in base 8, etc. As well in the other direction, real numbers in base 10 vs real numbers in base 11, and so on.

Basically, considering only natural numbers as bases, there are a countable infinity of uncountable infinities!

Answer 1

Basically all these sets are the same size:

• numbers with nines
• numbers without nines
• numbers expressed in base nine

That's how infinity works. For example, using a countable infinity, you can generate a one-to-one correspondence between the integers and numbers containing only the digit nine:

1      9
2     99
3    999
4   9999
5  99999
...
n  10ⁿ-1

If you were to take any N numbers, I think you would agree that those containing only nines are much rarer than the rest, so their frequency would approach 0%, even with a one-to-one correspondence with the integers.